WEBVTT
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GILBERT STRANG: OK.
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Well, the idea of
this first video
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is to tell you what's coming,
to give a kind of outline
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of what is reasonable to learn
about ordinary differential
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equations.
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And a big part of
the series will
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be videos on first order
equations and videos
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on second order equations.
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Those are the ones you
see most in applications.
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And those are the ones you
can understand and solve,
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when you're fortunate.
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So first order equations
means first derivatives
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come into the equation.
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So that's a nice equation
that we will solve,
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we'll spend a lot of time on.
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The derivative is-- that's
the rate of change of y--
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the changes in the unknown
y-- as time goes forward
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are partly from depending
on the solution itself.
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That's the idea of a
differential equation,
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that it connects the changes
with the function y as it is.
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And then you have
inputs called q of t,
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which produce their own change.
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They go into the system.
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They become part of y.
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And they grow, decay,
oscillate, whatever y of t does.
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So that is a linear equation
with a right-hand side,
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with an input, a forcing term.
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And here is a
nonlinear equation.
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The derivative of y.
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The slope depends on y.
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So it's a differential equation.
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But f of y could be y squared
over y cubed or the sine of y
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or the exponential of y.
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So it could be not linear.
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Linear means that
we see y by itself.
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Here we won't.
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Well, we'll come
pretty close to getting
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a solution, because it's
a first order equation.
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And the most general first
order equation, the function
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would depend on t and y.
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The input would
change with time.
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Here, the input depends only
on the current value of y.
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I might think of y
as money in a bank,
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growing, decaying, oscillating.
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Or I might think of y as
the distance on a spring.
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Lots of applications coming.
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OK.
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So those are first
order equations.
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And second order have
second derivatives.
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The second derivative
is the acceleration.
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It tells you about the
bending of the curve.
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If I have a graph, the
first derivative we know
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gives the slope of the graph.
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Is it going up?
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Is it going down?
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Is it a maximum?
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The second derivative tells
you the bending of the graph.
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How it goes away
from a straight line.
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So and that's acceleration.
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So Newton's law-- the
physics we all live with--
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would be acceleration
is some force.
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And there is a force that
depends, again, linearly--
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that's a keyword-- on y.
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Just y to the first power.
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And here is a little bit
more general equation.
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In Newton's law,
the acceleration
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is multiplied by the mass.
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So this includes a physical
constant here, the mass.
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Then there could
be some damping.
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If I have motion, there may
be friction slowing it down.
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That depends on the first
derivative, the velocity.
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And then there could be the
same kind of forced term
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that depends on y itself.
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And there could be some outside
force, some person or machine
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that's creating movement.
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An external forcing term.
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So that's a big equation.
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And let me just
say, at this point,
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we let things be nonlinear.
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And we had a pretty good chance.
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If we get these
to be non-linear,
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the chance at second
order has dropped.
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And the further
we go, the more we
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need linearity and maybe
even constant coefficients.
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m, b, and k.
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So that's really
the problem that we
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can solve as we get good at
it is a linear equation--
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second order, let's say--
with constant coefficients.
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But that's pretty
much pushing what
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we can hope to do
explicitly and really
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understand the
solution, because so
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linear with constant
coefficients.
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Say it again.
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That's the good equations.
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And I think of
solutions in two ways.
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If I have a really nice
function like a exponential.
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Exponentials are
the great functions
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of differential equations, the
great functions in this series.
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You'll see them over and over.
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Exponentials.
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Say f of t equals-- e to the t.
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Or e to the omega t.
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Or e to the i omega t.
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That i is the square
root of minus 1.
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In those cases, we will get
a similarly nice function
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for the solution.
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Those are the best.
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We get a function that we
know like exponentials.
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And we get solutions
that we know.
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Second best are we get some
function we don't especially
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know.
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In that case, the
solution probably
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involves an integral of
f, or two integrals of f.
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We have a formula for it.
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That formula includes
an integration
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that we would have to
do, either look it up
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or do it numerically.
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And then when we get to
completely non-linear
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functions, or we have
varying coefficients,
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then we're going
to go numerically.
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So really, the wide,
wide part of the subject
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ends up as numerical solutions.
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But you've got a
whole bunch of videos
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coming that have nice
functions and nice solutions.
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OK.
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So that's first order
and second order.
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Now there's more, because
a system doesn't usually
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consist of just a single
resistor or a single spring.
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In reality, we have
many equations.
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And we need to deal with those.
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So y is now a vector.
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y1, y2, to yn.
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n different unknowns.
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n different equations.
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That's n equation.
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So here that is
an n by n matrix.
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So it's first order.
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Constant coefficient.
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So we'll be able
to get somewhere.
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But it's a system of
n coupled equations.
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And so is this one with
a second derivative.
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Second derivative
of the solution.
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But again, y1 to yn.
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And we have a matrix,
usually a symmetric matrix
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there, we hope, multiplying y.
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So again, linear.
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Constant coefficients.
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But several equations at once.
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And that will bring in
the idea of eigenvalues
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and eigenvectors.
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Eigenvalues and eigenvectors
is a key bit of linear algebra
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that makes these
problems simple,
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because it turns
this coupled problem
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into n uncoupled problems.
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n first order equations that
we can solve separately.
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Or n second order equations
that we can solve separately.
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That's the goal with
matrices is to uncouple them.
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OK.
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And then really the big
reality of this subject
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is that solutions
are found numerically
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and very efficiently.
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And there's a lot to learn
about that, a lot to learn.
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And MATLAB is a
first-class package
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that gives you numerical
solutions with many options.
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One of the options
may be the favorite.
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ODE for ordinary
differential equations 4 5.
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And that is numbers 4, 5.
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Well, Cleve Moler, who
wrote the package MATLAB,
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is going to create a
series of parallel videos
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explaining the steps
toward numerical solution.
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Those steps begin with
a very simple method.
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Maybe I'll put the
creator's name down.
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Euler.
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So you can know that because
Euler was centuries ago,
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he didn't have a computer.
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But he had a simple
way of approximating.
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So Euler might be ODE 1.
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And now we've left Euler behind.
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Euler is fine, but not
sufficiently accurate.
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ODE 45, that 4 and 5 indicate a
much higher accuracy, much more
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flexibility in that package.
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So starting with
Euler, Cleve Moler
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will explain several
steps that reach
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a really workhorse package.
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So that's a parallel series
where you'll see the codes.
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This will be a
chalk and blackboard
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series, where I'll find
solutions in exponential form.
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And if I can, I would like to
conclude the series by reaching
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partial differential equations.
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So I'll just write some partial
differential equations here,
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so you know what they mean.
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And that's a goal
which I hope to reach.
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So one partial
differential equation
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would be du dt-- you see
partial derivatives-- is
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second derivative.
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So I have two variables now.
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Time, which I always have.
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And here is x in
the space direction.
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That's called the heat equation.
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That's a very important
constant coefficient,
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partial differential equation.
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So PDE, as distinct from ODE.
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And so I write down one more.
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The second derivative of u
is the same right-hand side
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second derivative
in the x direction.
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That would be called
the wave equation.
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So this is like the first
order equation in time.
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It's like a big system.
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In fact, it's like an infinite
size system of equations.
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First order in time.
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Or second order in time.
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Heat equation.
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Wave equation.
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And I would like to also
include a the Laplace equation.
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Well, if we get there.
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So those are goals for
the end of the series that
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go beyond some courses in ODEs.
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But the main goal
here is to give you
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the standard clear picture
of the basic differential
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equations that we can
solve and understand.
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Well, I hope it goes well.
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Thanks.